If you want to explain this in layman's terms I suggest to take a concrete example (e.g a driving car) and ask questions like: How would you describe the motion of the car quantitatively over the phone to somebody? This should naturally lead you to realise that you need to explain in which direction and how fast (magnitude) it is driving, I.e. you already arrive at the vector concept.
Similarly for force, think of an object you could push (accelerate) in different directions...
If you don't want equations, all you can do is thinking of how you would describe these quantities in words. For example you could say:
The car is driving at 100 km/h.
This is its speed (scalar). Note that you are not specifying the direction in which it is driving.
The car is driving at 100 km/h in the middle of the road to Paris.
This is velocity (vector). You specify the direction (in the middle of the road to Paris) and the magnitude (100 km/h). Mathematically, direction and magnitude together can be expressed by a vector ("arrow" if you want), where the length of the vector/arrow is a measure for the magnitude/speed.
As you can see from the example, "speed" includes the same information as "velocity" (minus the direction). So if you always use velocity vectors you can extract the speed information (=length of vector).
Acceleration is the rate of change of velocity of an object with respect to time, i.e. it describes how quickly the velocity changes. This can be both, a change of the magnitude (speed) of the velocity and/or a change of the direction of the velocity (even if the magnitude/speed stays the same).
In the same way as the velocity/speed, acceleration can be a scalar or a vector quantity depending on what you describe.
This car accelerates from 0 to 100 km/h in 10 s.
Here acceleration is a scalar (=10 km/ (h s)) as you describe the change of the speed of the car. You don't make any statements about direction.
In this turn the formula 1 car driver is subject to a lateral acceleration of $4 g$.
Here, acceleration is a vector as you specify the direction (lateral, i.e. to the side) and magnitude ($4 g$, i.e. four times the earth lateral acceleration).
Force (in these examples) is just acceleration (vector or scalar) multiplied by mass, so basically you can use exactly the same examples, replacing acceleration with force and multiplying the results with the mass of the car (first example) and the mass of the driver (second example).
Scalar / Vector in general
Many physical quantities can appear both as vector and scalar. If you always work with the vector version you can't go wrong, but at times this might contain more information than necessary. For instance to calculate the kinetic energy of the car you only need the magnitude/speed and don't need the direction of the velocity vector. As an example:
This car, when driving at 90 km/h uses 5 liters of petrol in 100 km.
This car, when driving at 90 km/h in the middle of the road to Paris uses 5 liters of petrol in 100 km.
As you can see, while the second (vector) version of the sentence is correct, it contains unnecessary superfluous information on the direction.
Usually there are no special words to distinguish vector from scalar quantities (unlike "speed/velocity"). You could use constructions like "force vector", "scalar force"... to make that distinction. Even for speed/velocity, in my experience this distinction is not consistently used. Personally I'd use "velocity" also for the scalar quantity "speed" and some people might conversely call everything (scalar and vector) by the name "speed".